Calculus Examples

$lim_{x \to - 8} 2 x + 1 - \sqrt{x^{2} - 1}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$lim_{x \to - 8} 2 x + lim_{x \to - 8} 1 - lim_{x \to - 8} \sqrt{x^{2} - 1}$

Step 2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to - 8} x + lim_{x \to - 8} 1 - lim_{x \to - 8} \sqrt{x^{2} - 1}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

$2 lim_{x \to - 8} x + 1 - lim_{x \to - 8} \sqrt{x^{2} - 1}$

Step 4

Move the limit under the radical sign.

$2 lim_{x \to - 8} x + 1 - \sqrt{lim_{x \to - 8} x^{2} - 1}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$2 lim_{x \to - 8} x + 1 - \sqrt{lim_{x \to - 8} x^{2} - lim_{x \to - 8} 1}$

Step 6

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$2 lim_{x \to - 8} x + 1 - \sqrt{{(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} 1}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

$2 lim_{x \to - 8} x + 1 - \sqrt{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 1}$

Step 8

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$2 \cdot - 8 + 1 - \sqrt{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 1}$

Step 8.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$2 \cdot - 8 + 1 - \sqrt{{(- 8)}^{2} - 1 \cdot 1}$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $2$ by $- 8$ .

$- 16 + 1 - \sqrt{{(- 8)}^{2} - 1 \cdot 1}$

Step 9.1.2

Raise $- 8$ to the power of $2$ .

$- 16 + 1 - \sqrt{64 - 1 \cdot 1}$

Step 9.1.3

Multiply $- 1$ by $1$ .

$- 16 + 1 - \sqrt{64 - 1}$

Step 9.1.4

Subtract $1$ from $64$ .

$- 16 + 1 - \sqrt{63}$

Step 9.1.5

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 9.1.5.1

Factor $9$ out of $63$ .

$- 16 + 1 - \sqrt{9 (7)}$

Step 9.1.5.2

Rewrite $9$ as $3^{2}$ .

$- 16 + 1 - \sqrt{3^{2} \cdot 7}$

Step 9.1.6

Pull terms out from under the radical.

$- 16 + 1 - (3 \sqrt{7})$

Step 9.1.7

Multiply $3$ by $- 1$ .

$- 16 + 1 - 3 \sqrt{7}$

Step 9.2

Add $- 16$ and $1$ .

$- 15 - 3 \sqrt{7}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 15 - 3 \sqrt{7}$

Decimal Form:

$- 22.93725393 \dots$